Mathbox for Richard Penner < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  elcnvcnvintab Structured version   Visualization version   GIF version

Theorem elcnvcnvintab 38407
 Description: Two ways of saying a set is an element of the converse of the converse of the intersection of a class. (Contributed by RP, 20-Aug-2020.)
Assertion
Ref Expression
elcnvcnvintab (𝐴 {𝑥𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑𝐴𝑥)))
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝜑(𝑥)

Proof of Theorem elcnvcnvintab
StepHypRef Expression
1 cnvcnv 5727 . . . 4 {𝑥𝜑} = ( {𝑥𝜑} ∩ (V × V))
2 incom 3954 . . . 4 ( {𝑥𝜑} ∩ (V × V)) = ((V × V) ∩ {𝑥𝜑})
31, 2eqtri 2792 . . 3 {𝑥𝜑} = ((V × V) ∩ {𝑥𝜑})
43eleq2i 2841 . 2 (𝐴 {𝑥𝜑} ↔ 𝐴 ∈ ((V × V) ∩ {𝑥𝜑}))
5 elinintab 38400 . 2 (𝐴 ∈ ((V × V) ∩ {𝑥𝜑}) ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑𝐴𝑥)))
64, 5bitri 264 1 (𝐴 {𝑥𝜑} ↔ (𝐴 ∈ (V × V) ∧ ∀𝑥(𝜑𝐴𝑥)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 382  ∀wal 1628   ∈ wcel 2144  {cab 2756  Vcvv 3349   ∩ cin 3720  ∩ cint 4609   × cxp 5247  ◡ccnv 5248 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ral 3065  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-int 4610  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-cnv 5257 This theorem is referenced by:  cnvcnvintabd  38425
 Copyright terms: Public domain W3C validator